Kelly Fraction: f* ≈ 10% — bet 10% of bankroll each round for maximum long-term growth
With 60% win probability and 2:1 payout, the Kelly criterion recommends betting 10% of your bankroll each round. This maximizes the expected logarithmic growth rate at approximately 0.5% per round. Over-betting (e.g., 50%) leads to dramatic volatility and frequent ruin, while under-betting (e.g., 2%) grows safely but slowly.
What Is the Kelly Criterion?
The Kelly criterion is a formula that determines the optimal fraction of your bankroll to wager on a favorable bet. Discovered by John L. Kelly Jr. at Bell Labs in 1956, it was originally developed in the context of information theory and noisy communication channels. The key insight is that maximizing the expected logarithm of wealth — rather than expected wealth itself — leads to the fastest possible long-term growth rate.
The Mathematics of Optimal Betting
For a simple bet with win probability p, win payout b, and loss amount a, the Kelly fraction is f* = p/b - q/a, where q = 1-p. This elegant formula balances greed (betting large to exploit your edge) against fear (betting small to survive bad streaks). The resulting growth rate g = p·ln(1+f·b) + q·ln(1-f·a) is maximized at exactly the Kelly fraction.
Over-Betting and the Path to Ruin
The simulation vividly demonstrates why over-betting is dangerous. Watch the red trajectories (2× Kelly) — they swing wildly and frequently crash to near-zero. The cyan Kelly trajectories grow steadily, while the blue under-bet paths are safe but slow. The all-in strategy (gray) almost certainly hits zero. This illustrates a deep mathematical truth: the arithmetic average of outcomes can be positive while the geometric average (which determines compound growth) is negative.
From Gambling to Investing
The Kelly criterion has found wide application beyond gambling. Warren Buffett, Bill Gross, and Jim Simons have all cited Kelly-like reasoning in their investment approaches. Edward Thorp used it to beat casinos at blackjack and later to manage a hugely successful hedge fund. In practice, most professionals use fractional Kelly (half or quarter Kelly) to reduce volatility and account for uncertainty in their edge estimates.
FAQ
What is the Kelly criterion?
The Kelly criterion is a formula for determining the optimal size of a series of bets to maximize long-term wealth growth. Developed by John Kelly at Bell Labs in 1956, it states that you should bet a fraction f* = p/b - q/a of your bankroll, where p is win probability, q = 1-p, b is the net payout on a win, and a is the net loss on a loss.
Why does over-betting lead to ruin?
Over-betting (betting more than Kelly) increases variance faster than it increases expected return. While the arithmetic mean of outcomes may be higher, the geometric mean (which determines long-term compound growth) actually decreases. In the extreme, betting 100% of your bankroll means a single loss wipes you out entirely.
What is fractional Kelly?
Fractional Kelly means betting a fraction (typically half) of the full Kelly amount. This reduces the expected growth rate only slightly while dramatically reducing volatility and drawdowns. Many professional investors and gamblers use half-Kelly or quarter-Kelly in practice to account for estimation errors in their edge.
How does the Kelly criterion relate to portfolio management?
The Kelly criterion is mathematically equivalent to maximizing expected logarithmic utility of wealth. In portfolio theory, this corresponds to the growth-optimal portfolio. While Markowitz's mean-variance framework minimizes risk for a given return, Kelly maximizes the long-run compound growth rate, making it especially relevant for repeated investment decisions.